The solution of ordinary differential equations with constant coefficients by means of the L-transformation is accomplished very easily, since the L-transformation removes the differentiation, a transcendental operation, and an algebraic image equation is obtained. When the original equation contains derivatives with respect to two variables, say x and t, that is it represents a partial differential equation, then the L-transformation applied to the variable t will remove differentiation with respect to t, and the image equation is an ordinary differential equation, with the variable x. Obviously, for this purpose we must presume that t varies in the interval 0 ≦ t < ∞ the variable x may range in an interval which may be bounded or unbounded at one or both sides. Accordingly, we have in the xt-plane (see Fig. 48) as fundamental region of the partial differential equation either a half-strip or a quadrant or a half-plane.


Partial Differential Equation Ordinary Differential Equation Singular Point Asymptotic Expansion Image Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Gustav Doetsch
    • 1
  1. 1.Emeritus of MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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